Analysis of the latitudinal data of Eratosthenes and
Hipparchus
Christian Marx∗
Abstract: The handed down latitudinal data ascribed to Eratosthenes and Hipparchus
are composed and each tested for consistency by means of adjustment theory. For
detected inconsistencies new explanations are given concerning the origin of the data.
Several inconsistent data can be ascribed to Strabo. Differences in Hipparchus’ data
can often be explained by the different types and precision of the data. Gross errors in Eratosthenes’ data are explained by their origination from the lengths of sea
routes. From Eratosthenes’ data concerning Thule a numerical value for Eratosthenes’
obliquity of the ecliptic is deduced.
1
Introduction
A precise specification of positions on the earth surface became possible in ancient
geography by the introduction of reference systems and physical quantities for the
description of positions. Eratosthenes (ca. 276–194 BC), founder of mathematical
geography, introduced a grid of non-equidistant parallels and meridians for the positions of selected places. In his “Geography” he described the inhabited world by
means of distance data and expressed his latitudinal data probably by means of meridian arc lengths. The astronomer and mathematician Hipparchus (ca. 190–120 BC)
probably introduced the division of the full circle into 360◦ into Greek astronomy
and geography (e.g., [Dicks, 1960, p. 149]). He transferred the concept of ecliptical
longitude and latitude for the specification of star positions to the terrestrial sphere.
Hipparchus’ essential geographical work is his treatise “Against the ‘Geography’ of
Eratosthenes”, wherein he discussed the works of Eratosthenes and gave a compilation of latitudes and equivalent astronomical quantities for several locations. Later
Ptolemy (ca. 100–170 AD) used Hipparchus’ concept and introduced a geographical
coordinate system for his position data in his “Geography” (Geographike Hyphegesis,
GH), which differs from today’s system only by its zero meridian.
The mentioned works of Eratosthenes and Hipparchus are handed down only in
fragments, mainly by Strabo’s (ca. 63 BC – 23 AD) “Geography” (G; see [Jones, 1932],
[Radt, 2011]). The geographical fragments of Eratosthenes and Hipparchus were compiled and commented on by [Berger, 1880] and [Roller, 2010] as well as by
[Berger, 1869] and [Dicks, 1960], respectively. In particular the latitudinal data of
the fragments are given partly redundantly and with differing numerical values. It is
uncertain, whether all the data originate from Eratosthenes or Hipparchus, respectively (see also [Roller, 2010, p. 36]). Therefore, an investigation of their consistency is
indicated. The aim of this contribution is to carry out such an investigation conjointly
∗ Christian
Marx, Gropiusstraße 6, 13357 Berlin, Germany; e-mail: ch.marx@gmx.net.
Revision submitted to MEMOCS on 11 August 2014.
1
for all considered data so that it is ensured and shown that all relations between the
data are integrated. For this purpose, the data are composed in systems of equations;
when solving these systems appropriately, the inconsistencies of the data become evident (Sections 2, 3). For inconsistencies new explanations are given. The actual
accuracy of the investigated ancient data is not subject of this contribution. Some
grossly erroneous data of Eratosthenes, however, are explained by their origination
from the lengths of sea routes (Section 2.3).
Among Eratosthenes’ latitudinal data there is a distance concerning Thule, the
place visited by the geographer and astronomer Pytheas during his expedition to
Great Britain in about 330 BC. From Eratosthenes’ and Pytheas’ information concerning Thule a numerical value for Eratosthenes’ obliquity of the ecliptic is deduced
(Section 2.4). A localisation of Thule based on new considerations of Pytheas’ sea
route and of the lengths of the nights in Thule is to be found in the appendix.
2
Eratosthenes’ latitudinal data
The latitudinal data of Eratosthenes considered in the following are based on the
fragments given by [Roller, 2010] (the following translations are taken from it). The
investigations are limited to the locations in the western Oikoumene (the inhabited
world known to the Greeks and Romans), in particular those in connection with
Eratosthenes’ prime meridian through Rhodes, because only these data are partly redundant. The considered data originate from Strabo’s “Geography”, Pliny’s “Natural
History” (NH; see [Bostock and Riley, 1855]) and Cleomedes’ “Caelestia” (C). Figure
1 shows some of the locations.
Strabo and possibly Eratosthenes expressed latitudes and latitudinal differences
by means of meridian arc lengths (b hereinafter and b0 if with respect to the equator)
in the measurement unit stadium (st). Eratosthenes introduced the value
C = 252,000 st
(1)
for the circumference of the earth (e.g., G II.5.7) so that for a meridian arc holds true
1◦ =
b C/360 = 700 st .
(2)
The considered information presumably originating from Eratosthenes and the
corresponding fragments (F) and sources are given in Table 1.1 The F-numbers
correspond to [Roller, 2010]. The data occuring repeatedly within one fragment are
listed and used once only. In addition, consecutive numbers were introduced, which
are separated from the F-number by a dot. If it follows from the textual source that
two locations have the same latitude, b is set to 0. Figure 2 visualises the data in a
graph. An edge is set between two locations if at least one b exists for them.
2.1
Notes on the data
The southern limit of the inhabited world is constituted by the Cinnamon country
(F34/G II.5.7). For the northern limit, Eratosthenes gives two locations of differing
latitudes: the “northern regions” (F34/G II.5.9) and Thule (F35/G I.4.2).
1 The
symbols . and & stand for “somewhat smaller” and “somewhat larger”.
2
The Cinnamon country corresponds to the stretch of coast between Cape Guardafui and the straits of Bab-el-Mandeb (cf. [Dicks, 1960, p. 171]). The information of
F30.1, F30.2 (G II.5.6) and F34.11 (G II.5.7) does not refer directly to the Cinnamon
country but to the southern limit of the inhabited world. Their identicalness results
from G II.5.7.
The data on the Borysthenes, the Dnieper River2 , refer to its mouth into the Black
Sea.
The latitude of the Hellespont (Dardanelles) presumably corresponds to that of
Eratosthenes’ parallel through Lysimachia (near Bolayır, Gallipoli peninsula), which
is mentioned in G II.5.40 (likewise [Berger, 1880, p. 155]).
4
F34 (G II.5.7): Strabo says that b0 of the tropic of Cancer corresponds to 60
of
C (i.e. C/15 = 16,800 st), that the tropic goes through Syene and that b0 of Syene is
1
16,800 st. The latter is applied here (F34.5). Eratosthenes uses 15
of the full circle
for the obliquity of the ecliptic ε in this case, i.e.
εr = 24◦ ,
(3)
which was a common value in early Greek geography (cf. [Neugebauer, 1975, pp.
733–734]).
F34.13 (G II.5.8): This fragment is introduced in addition to [Roller, 2010] (see
Section 2.2) but not used in the following test of consistency.
F36.1, F36.2 (G II.5.42), F47.1 (G II.1.3): In G II.5.42 Strabo deals with Hipparchus’ data on the regions in the neighbourhood of the Borysthenes and the southern parts of Lake Maeotis (Sea of Azov) and he states: “Eratosthenes says that these
regions are a little more than 23,000 stadia [F36.1] from Meroë, since it is 18,000 stadia [F36.2] to the Hellespont and then 5,000 [F36.3] to Borysthenes.” [Roller, 2010,
p. 155] derives that the “. . . mouth of the Borysthenes is somewhat over 23,000 stadia from Meroë.” From the text, however, follows that b of Meroë – Borysthenes
is (18,000 + 5,000) st = 23,000 st. Thus, the text suggests that Eratosthenes differentiated between the latitude of the mouth of the Borysthenes and the latitude of
the mentioned regions, which are situated further north than the mouth.3 Another
interpretation results from F34.1/35.1 (Meroë – Alexandria) and F35.2 (Alexandria
– Hellespont), which yield b = 18,100 st for Meroë – Hellespont so that b of Meroë –
Borysthenes is 23,100 st. Possibly, Eratosthenes specified b = 23,100 st for the mouth
of the Borysthenes and its neighbouring regions and Strabo described this value by
“a little more than 23,000 st” and erroneously used 18,000 st instead of 18,100 st in his
statement. Moreover, in G II.1.3 Strabo states: “From Meroë to the Hellespont is no
more than 18,000 stadia [F47.1] . . . ”.4 Since, however, he compares this value with
the distance from India to the Bactrians, it is certainly only a rough value. Owing to
the differences in the information, the values 23,000 st and 18,000 st are not used in
the following calculational test of consistency.
F40 (G II.1.20), F41 (NH II.185): According to F40, Eratosthenes agrees closely
with Philo (Ptolemaic officer, see [Roller, 2010, p. 157]) that in Meroë the sun is at
2 In antiquity, there was disagreement on the location of the Borysthenes, cf. Pliny, NH IV.83.
For instance, Ptolemy (GH III.5) probably confuses the Borysthenes with the Hypanis and locates
the Borysthenes at the Southern Bug (cf. [Marx and Kleineberg, 2012, pp. 50, 53]).
3 In fact, however, the southern parts of Lake Maeotis are further south than the Borysthenes.
4 I thank one of the referees for his reference to this text passage.
3
the zenith 45 days before the summer solstice. In order to test this, the dates of
the year were determined for 350–250 BC when the sun altitude a was maximal in
Meroë (φ = 16◦ 56′ ).5 As a result, the sun reached its maximal a ≈ 90◦ 45 or 46
days before as well as 46 or 45 days after the day of the summer solstice. Hence, the
information of F40 is probably based on an accurate observation. Eratosthenes’ bdata yield b0 = 11,800 st for Meroë (see Section 2.2), which corresponds to φ = 16◦ 51′
in good agreement with the actual φ. According to F41, the shadows fall to the south
45 days before and after the summer solstice in the country of the Trogodytes. That
corresponds to the information of F40 on Meroë. It is not known whether Eratosthenes
derived latitudes from the information of F40 and F41. At least, however, it can be
assumed that Eratosthenes believed the Trogodytes and Meroë to be located at the
same latitude. Thus, b = 0 is introduced here for Meroe – Trogodytes only.
F128.1 (G II.5.24): The distance Alexandria – Rhodes is not explicitly indicated
as a latitudinal difference, but it was found by “. . . using the shadow of a gnomon
. . . ” so that it is considered to be a b-value.
FM6 (C I.7): This fragment originates from Eratosthenes’ work “On the Measurement of the Earth”, see [Roller, 2010, pp. 263–267]. According to this, Syene is
located at the tropic of Cancer; thus, following F34, b0 = 16800 st is applied here.
2.2
Test of consistency
A test of the consistency of the data attributed to Eratosthenes is carried out simultaneously for all data by means of a formation of an equation system for the data.
j,k
For each given latitudinal data bj,k
= bk0 − bj0 is introduced, where
i , the equation bi
j,k
bi is the i-th given meridian arc length between the parallels of the j-th and k-th
location and bj0 and bk0 are the meridian arc lengths between these locations and the
equator. The latter are the unknown quantities of the equation system, they refer to
the 11 locations in Figure 2. If a j refers to the equator, bj0 is not an unknown but
has the value 0. If a bj,k
is specified by an inequality b > x or b < x, it is replaced by
i
b = x for the computation.
Since there are redundant data which are inconsistent among each other, the
numerical solution of the equation system is not directly possible. In order to achieve
a consistent equation system, an unknown correction vi (residual) is introduced for
each data bj,k
i :
j
k
bj,k
i = 1...n
(4)
i + vi = b0 − b0 ,
(n is the number of data), as it is usual in adjustment theory. System (4) can be
solved by means of the minimisation of an object function S of the corrections vi . In
5 The
time of the summer solstice can be determined according to [Meeus, 1991, pp. 165–167].
For a location of longitude λ and latitude φ and for a given time t, the sun altitude a can be
determined by the following calculation steps (formulas see [Meeus, 1991, pp. 84, 88–89, 135, 151–
3]): obliquity of the ecliptic: ε(t); mean anomaly of the sun: M (t); mean ecliptic longitude of the
sun: L0 (t); equation of center: C(t,M ); true ecliptic longitude: Θ(L0 , C); right ascension of the
sun: α(Θ, ε); declination of the sun: δ(Θ, ε); GMST: θ0 (t); hour angle: H(λ, α, θ0 ); a(φ, δ, H). (The
software-implemented calculation was tested by means of a comparison with the results of the online
calculator by [Cornwall et al., 2013].)
4
the present case the object function
S=
n
X
|vi | → min
(5)
i=1
of the L1 -norm adjustment is appropriate because it is a resistant estimation method
and therefore able to reveal inconsistencies of the data (see, e.g., [Marx, 2013a]; data
errors are manifested in large values of the vi ). For the b-values which occur multiply
in Table 1, multiple equations are introduced so that their influence is increased in
the adjustment. The L1 -norm adjustment is numerically solved here by the simplex
algorithm by [Barrodale and Roberts, 1974] (BR-algorithm).
The imprecise data of F30.1, F34.12 and F56.1 expressed by inequalities are not
included in the adjustment but also obtain corrections vi based on the determined
unknowns. There remain n = 22 data for the adjustment. The solution of the BRalgorithm yields 18 vi being 0; the belonging b-data are therefore consistent among
each other. The other vi 6= 0 are considered in the following. It shows that the
solution for the b0 can be regarded as to be in accord with Eratosthenes’ original
data, with the exception of the “northern regions” (see below). The b0 are given in
Figure 2.
F30.1, F34.12; Cinnamon country – “northern regions”; b < 30,000 st; v = 100 st:
b < 30,000 st is contradictory to v > 0. If this is Eratosthenes’ information, probably
not all b-data of F34.3, F34.1/35.1, F35.2, F35.3/36.3 and F34.9 (Cinnamon country
– Meroë – Alexandria – Hellespont – Borysthenes – “northern regions”) originate from
Eratosthenes because they yield 30,100 st. One explanation is that b = 4000 st for
Borysthenes – “northern regions” (F34.9/G II.5.9) is not from Eratosthenes but from
Strabo. This b-value is already given in the preceding section G II.5.8: “For, so far
as science is concerned, it is sufficient to assume that, just as it was appropriate in
the case of the southern regions to fix a limit of the habitable world by proceeding
three thousand stadia south of Meroë [. . . ], so in this case too we must reckon not
more than three thousand stadia [F34.13] north of Britain [i.e. Borysthenes], or only
a little more, say, four thousand stadia” ([Jones, 1932, p. 445]). (The latitude of
Britain equals that of the Borysthenes according to G II.5.8.) The text suggests that
the mentioned value of 3,000 st (F34.13) may originate from Eratosthenes and the
value of 4,000 st may be an alteration by Strabo. By means of 3,000 st, the considered
extent of the inhabited world is 29,000 st, which fulfills the condition b < 30,000 st of
F30.1 and F34.12, and b0 of the “northern regions” is 37,900 st.
F34.10; Rhodes – “northern regions”; b = 12,700 st; v = 650 st: b is contradictory to the value 13,350 st which follows from F128.1 (Alexandria – Rhodes) and
F35.2, F35.3/36.3, F34.9 (Alexandria – Hellespont – Borysthenes – “northern regions”). Possibly, b = 12,700 st originates from Strabo. 12,700 st less 4,000 st for the
part Borysthenes – “northern regions” (F34.9) yields 8700 st for the part Rhodes –
Borysthenes, but F128.1, F35.2 and F35.3/36.3 yield 9,350 st. The value 8,700 st,
however, nearly corresponds to Hipparchus’ value 8,600 st (cf. G II.1.12, II.5.41) so
that Strabo possibly chose it following Hipparchus and used it for b of F34.10.
F34.11; Cinnamon country – Rhodes; b = 16,600 st; v = 150 st: b is contradictory
to the sum 16,750 st of F34.3, F34.1/35.1, F128.1 (Cinnamon country – Meroë –
Alexandria – Rhodes). Possibly, b = 16,600 st is Strabo’s sum, which is not based on
5
3,750 st for the part Alexandria – Rhodes (F128.1) but on 3,600 st which is given by
Hipparchus (cf. G II.5.39 and Section 3.2).
F35.5; Cinnamon country – Meroë; b = 3,400 st; v = −400 st: b is contradictory
to b = 3,000 st of F34.3, which equals Hipparchus’ b in G II.5.35. Strabo says in F35:
“. . . if we add 3,400 [F35.5] more beyond Meroë, so that we include the Egyptian
island, the Kinnamomophoroi [Cinnamon country], and Taprobane, we have 38,000
stadia [F35.6].” Hence, the reason for an alteration of b by Strabo could be the
extent of the Egyptian island and/or of Taprobane. In F53 (G II.5.14) Strabo states
that the Cinnamon country, Taprobane and the Island of the Egyptians are situated
on the same parallel, but in view of the spatial extent of these three locations this
can only be an approximate piece of information (e.g., Eratosthenes’ estimate of
the latitudinal extent of Taprobane is 7,000 st according to F76/NH VI.81). Strabo
possibly introduced 3,400 st in order to obtain the round value of 38,000 st (F35.6)
for the latitudinal extent of the inhabited world from its southern limit to Thule.
F35.6; Cinnamon country – Thule; b = 38,000 st; v = −400 st: b equals the sum
of the other b-data of F35 but it is 400 st too large with regard to the sum based
on Eratosthenes’ presumable b = 3,000 st (F34.3) for Cinnamon country – Meroë,
see F35.5. Thus, b = 38,000 st is probably only a round sum for the extent of the
inhabited world given by Strabo.
F36.1; Meroë – Borysthenes; b & 23,000 st; v = 100 st: Since the information on b
is consistent with v > 0 and the small v, F36.1 can be regarded as consistent.
F36.2, F47.1; Meroë – Hellespont; b = 18,000 st; v = 100 st: b is contradictory to
the sum b = 18,100 st of F34.1/35.1 and F35.2 (Meroë – Alexandria – Hellespont).
Strabo ascribes F35.1 and F35.2 explicitly to Eratosthenes. The value 18,000 st possibly originates from Strabo, see Section 2.1.
F56.1; Alexandria – Rhodes; b . 4,000 st; v = −250 st: The information on b is in
accord with v < 0 and v is acceptable because 4,000 st is probably a roughly rounded
value; hence, F56.1 can be considered to be consistent.
2.3
Sea routes in Eratosthenes’ latitudinal data
The two southernmost b of Cinnamon country – Meroë (F34.3/35.5) and of Meroë –
Alexandria (F35.1) are (almost) correct. b = 3,000 st of F34.3 and b = 3,400 st
of F35.5 are converted by means of (2) 4◦ 17′ and 4◦ 51′ , respectively, and actually
4◦ 43′ (based on the central latitude 12◦ 13′ for the Cinnamon country). b of F35.1 is
converted as well as actually 14◦ 17′ .
In contrast, the subsequent latitudinal differences Alexandria – Hellespont of F35.2
and Hellespont – Borysthenes of F35.3/36.3 show large errors. b = 8,100 st of F35.2
is 11◦ 34′ and actually 9◦ 23′ ; the error is 2◦ 11′ =
b 1,528 st. b = 5,000 st of F35.3/36.3
is 7◦ 09′ and actually 6◦ 01′ ; the error is 1◦ 08′ =
b 793 st.
Both erroneous latitudinal differences are explicable by Eratosthenes’ conception
that the prime meridian through Rhodes also runs through Meroë, Alexandria, Caria,
Byzantium and (the mouth of) the Borysthenes (cf. G I.4.1, II.1.12, II.1.40). Figure 1
shows the position of the concerning locations. Moreover, Strabo says that it is generally agreed that the sea route Alexandria – Borysthenes is a straight line (G II.5.7).
Consequently, it is likely that the latitudinal differences Alexandria – Hellespont and
Hellespont – Borysthenes are based on the lengths of the sea routes, which were sup-
6
posed to take course along the meridian (also assumed by [Bunbury, 1879, p. 640],
[Roller, 2010, p. 152]). This is considered more in detail in the following. The lengths
of the sea routes could have been derived from journey times and estimates of the
speed, which was a usual procedure according to GH I.9.4, I.17.6.
F35.2, Alexandria – Hellespont: At least from its part Alexandria – Rhodes it is
known that Eratosthenes had information on the length of the sea route from navigators. For this part Eratosthenes gives his own b = 3,750 st (F128.1) and additionally
the two lengths 4,000 st and 5,000 st for the corresponding sea route based on the assumptions of navigators (F128/G II.5.24). The large difference between both lengths
shows the large uncertainty of such information. If for the latitudinal difference
Alexandria – Rhodes Eratosthenes’ 3,750 st are used, then (8,100−3,750) st = 4,350 st
remain for the rest of distance F35.2, i.e. for Rhodes – Hellespont. Figure 1 shows
a possible sea route from Rhodes to Hellespont (Lysimachia). It has a length of ca.
650 km. In order to convert this into stadia, not (2) is used (which only applies to b)
but a conventional stadium length. The Egyptian stadium of 157.5 m (cf. [Dilke, 1985,
p. 33]) is chosen6 , which yields 650 km = 4,127 st in good agreement with the ancient
value of 4,350 st.
F35.3/36.3, Hellespont – Borysthenes: b of the part Byzantium – Borysthenes is
3,800 st according to F34.8 (G II.5.8; this value is ca. 100 st less than the actual value).
For the rest of distance F35.3, i.e. for Hellespont – Byzantium, (5,000 − 3,800) st =
1,200 st remain. The assumed sea route Hellespont – Byzantium shown in Figure 1
has a length of 195 km = 1,238 st in good agreement with the ancient value of 1,200 st.
2.4
Eratosthenes’ obliquity of the ecliptic
Strabo states (F34/G II.5.8) that according to Pytheas in Thule the arctic circle
coincidences with the tropic of Cancer. The arctic circle delimits the region of the
circumpolar stars in the sky, which do not set (see, e.g., [Dicks, 1960, p. 165]). Thus,
its declination is
δa = 90◦ − φ ,
(6)
where φ is the latitude of the observer. Strabo’s information means that δa equals the
obliquity of the ecliptic ε. Hence, for φ of Thule φ = 90◦ − ε holds true so that Thule
is situated at the northern polar circle. There the sun does not set at the summer
solstice, which corresponds to Pliny’s information (NH IV.30) that at the summer
solstice there are no nights in Thule. ε was 23◦ 44′ at the time of Pytheas’ voyage7 so
that the northern polar circle was at φ = 66◦ 16′ . At Eratosthenes’ time ε was 23◦ 43′
so that φ = 66◦ 17′ .
6 For the Egyptian stadium there is some evidence. Strabo says (G X.4.5) that the sea voyage from
Cyrenaea to Criumetopon/Cape Krio (cf. [Bostock and Riley, 1855, IV.20, fn. 4]) in Crete takes two
days and nights and that Eratosthenes gave 2,000 st for this distance. Pliny also treats that distance
(NH IV.20) and gives more precisely Phycus/Cape Rasat (cf. [Bostock and Riley, 1855, IV.20, fn.
20]) instead of Cyrenaea. The distance Cape Rasat – Cape Krio is 313 km, which yields 156.5 km per
day. Eratosthenes’ distance is probably based on the definition: 1 day-and-night-seafaring =
b 1,000 st,
which was also assessed by Theophilus and Marinus (GH I.9.4). Hence, journey times were converted
into distances by means of this definition; from it and the distance Cape Rasat – Cape Krio follows
1 st ≈ 156.5 m. This is in accordance with the Egyptian stadium of 157.5 m so that this stadium
possibly underlies the aforesaid definition.
7 Computed according to [Meeus, 1991, p. 135].
7
The computation of Section 2.2 yields b0 = 46,400 st for Thule. This result is
composed of the following b-data:
1. equator – Cinnamon Country: 8,800 st (F30.2, F34.6);
2. Cinnamon Country – Meroë: 3,000 st (F34.3, the value occurs twice in F34);
3. Meroë – Alexandria: 10,000 st (F34.1, F35.1);
4. Alexandria – Hellespont: 8,100 st (F35.2);
5. Hellespont – Borysthenes: 5,000 st (F35.3, F36.3);
6. Borysthenes – Thule: 11,500 st (F35.4).
The 2nd b-value equals the difference between F34.4 (Cinnamon country – Syene) and
F34.2 (Meroë – Syene). The sum of 13,000 st of the 2nd and 3rd b-value (i.e. Cinnamon
country – Alexandria) equals the difference between F34.7 (Alexandria) and the 1st
b-value. The sum of the 3rd, 4th and 5th b-value (i.e. Meroë – Borysthenes) amounts
to 23,100 st and is confirmed by F36.1, where & 23,000 st is given. Hence, considering
the 3rd and 5th b-value to be correct, also the 4th b-value is confirmed. The 6th
b-value was probably calculated from the b0 -values of Thule and the Borysthenes.
◦ ′
Eratosthenes’ b0 of Thule corresponds to 46,400 st/700 st
◦ ≈ 66 17 , which equals
the actual position of the polar circle at Eratosthenes’ time. Apparently, he had
a good knowledge of the value of ε, which he used in conjunction with Pytheas’
information for the localisation of Thule. Assuming for b0 = 46,400 st a resolution of
100 st, the limits for ε are:
90◦ − (46,450/700)◦ < ε < 90◦ − (46,350/700)◦
(7)
23◦ 39′ < ε < 23◦ 47′ .
(8)
Ptolemy states in his Mathematike Syntaxis (MS; see [Manitius, 1912],
[Toomer, 1984]) I.12 that the ratio t = 11
83 of the arc between the tropics to the
full meridian equals nearly Eratosthenes’ value, which was also used by Hipparchus.
Ptolemy’s t leads to
εm = 23◦ 51′ 20′′ .8
(9)
Hipparchus presumably used
εh = 23◦ 40′
(10)
(cf. [Diller, 1934]). Probably, this is Eratosthenes’ value. It corresponds to t ≈ 10.91
83 ,
which does not differ significantly from Ptolemy’s value. For the polar circle, i.e.
Thule, it yields φ = 66◦ 20′ and b0 = 46,433 st ≈ 46,400 st in agreement with the value
resulting from Eratosthenes’ b-data. Possibly Eratosthenes specified b of Borysthenes
– Thule as “about” 11,500 st because it was calculated from b0 = 46,433 of Thule and
b0 = 34,900 st of the Borysthenes so that 11,533 st was obtained. Or he considered
the derived b of Borysthenes – Thule to be unreliable because b0 of the Borysthenes
was based on the lengths of the sea routes reported by navigators (cf. Section 2.3).
4
From F34 (G II.5.7) can be derived that Eratosthenes’ value for ε was 60
of C (i.e.
◦
◦ ′
24 ). This value is not contradictory to 23 40 because it is based on the division of
the full circle into 60 parts. It was a common value for ε in early Greek geography,
and Eratosthenes probably gave this rough value as well as a more precise value in
his works. Later ancient authors also mention or use this value, although a more
8 [Manitius, 1912, vol. 1, p. 44, fn. b] wrongly infers from MS I.12 that ε was Eratosthenes’ value.
m
According to the text, however, this applies only approximately (see also [Jones, 2011, p. 459]).
8
accurate value was known (e.g. Ptolemy in GH VII.6.7; see [Neugebauer, 1975, p.
734]). Strabo was probably not interested in Eratosthenes’ more precise value so that
4
C only. Similarly, in G II.5.43 Strabo refers the reader to
he adopted the value 60
Hipparchus’ work concerning astronomical matters.
3
Hipparchus’ latitudinal data
The investigations of Hipparchus’ latitudinal data are mainly based on the fragments
(F) given by [Dicks, 1960] (the following translations are taken from it). The data
mainly originate from Hipparchus’ treatise “Against the ‘Geography’ of Eratosthenes”,
which consisted of three books (cf. [Dicks, 1960, p. 37]). Latitudinal data occurred
in the second book (F12–34) and the third book (F35–63), the majority in the third
book. The third book contained astronomical data for several latitudes. Strabo gives
extracts of this compilation; for instance he limits the data to the inhabited world (cf.
G II.5.34). The occurring types of latitudinal data are: meridian arc length b or b0
between the parallels of two locations or from the equator (in st); noon altitude a of
the sun at the winter solstice given in astronomical cubits (c; 1 c = 2◦ ); ratio r = g : s
of the length g of the gnomon to the length s of its shadow; length M of the longest
day (summer solstice) in (equinoctial) hours. Presumably, the meridian arc lengths
do not originate from Hipparchus but were calculated by Strabo from latitudes by
means of relation (2) (e.g., [Berger, 1869, p. 37]). The M -data are compared with their
corresponding meridian arc lengths by [Rawlins, 2009]; for the sake of completeness,
however, they are included in the following investigation.
With the exception of F15 (G II.1.12), the latitudinal data in the fragments of Hipparchus’ second book (F19/G II.1.12, F22/G II.1.29, F24/G II.1.34, F26/G II.1.36) do
not have connections to the data in Hipparchus’ third book. The only data which positions the concerned locations absolutely in latitude is the imprecise information that
b of Athens – Babylon “. . . is not greater than 2,400 stades . . . ” (F22). Hipparchus
gives this limit only in order to show that Eratosthenes’ positioning of the Taurus is
wrong. Furthermore, there are no connections among the data of the second book
which would cause redundant relations among each other. Thus, these data are not
included in the following investigations; for a discussion of the data see [Dicks, 1960].
In places Strabo gives one latitudinal data which applies to several locations.
Among these locations there may be additions by Strabon which do not originate
from Hipparchus (see [Berger, 1869, p. 41]). Owing to uncertainties in this regard,
however, all locations are taken into account here. If within a fragment more than
two locations are related by one data, the derivable relations are kept as compact as
possible9 . The data which occur repeatedly within one fragment are listed and used
once only.
A further source is Hipparchus’ “Commentary on the Phenomena of Aratus and
Eudoxus” (CP; see [Manitius, 1894]), which contains only a few latitudinal data. In
the “Commentary” altitudes of the pole ap are given, which equal the latitude φ, as
9 In the case of information such as “A, B, C are x st distant from D, E, F”, not all nine derivable
distances from A, B, C to D, E, F are introduced but only the following five distances: A – D:
b = x st; A – B, A – C, D – E, D – F: b = 0. This is advisable because the value x was surely not
determined for all nine distances.
9
well as polar distances ζa of the ever visible circle (the arctic circle) or of the never
visible circle (which delimits the region of the stars which do not rise).
The considered data and their sources are composed in Table 2.10 The fragment
numbers 15–61 correspond to [Dicks, 1960] and are extended by a consecutive number
separated by a dot. F62–71 refer to the “Commentary” and are additionally introduced here (only partly mentioned by [Dicks, 1960], [Berger, 1869, p. 54, F V 11],
[Shcheglov, 2007]). Figure 3 shows the data in form of a graph.
For a comparison and a joint analysis of the consistency of the data, data not
given as b were converted into b. The conversions of the given quantities into φ are
considered in the following; φ can be converted into b0 by means of (2). If further
parameters are included in a conversion, it needs be considered for their choice whether
the quantities to be compared were originally independently determined or not.
The conversion of a into φ is
φ = 90◦ − a + δ ,
(11)
where δ is the declination of the sun.11 If the sun altitude as /aw refers to the summer/winter solstice, δ equals ε/−ε:
φ = 90◦ − as + ε
φ = 90◦ − aw − ε
(12)
(13)
(for the case of as /aw see Figure 4(a)/(b)). Since an ancient conversion from φ to a
is assumed here, for ε the value of εh (see (10)) is used which presumably underlies
Hipparchus’ conversion from M to φ (see [Diller, 1934]).
The ratio r refers to the equinox or the summer solstice. In the case of the equinox,
φ is computed from ratio re by
φ = arctan (1/re ) .
(14)
In the case of an equinoctial ratio, it is to be expected that it is not the result of
a measurement because at the equinox only unreliable gnomon measurements are
possible in contrast to the solstices (cf. [Rawlins, 2009]). In the case of a ratio rs
referring to the summer solstice, a real measurement can be expected and it holds
true that
φ = arctan (1/rs ) + ε .
(15)
In order to compare rs with an independently determined meridian arc length, for ε the
actual value must be used. ε = 23◦ 43′ of Hipparchus’ time is used here. Furthermore,
since the shadow is generated by the upper edge of the sun and not by its centre,
φ(rs ) must be enlarged by a systematic error of 16′ , whereby the radius of the sun
disc is taken into account (cf., e.g., [Dicks, 1960, p. 178]).
10 I thank one of the referees for the information that the value b = 12,500 st in G II.1.18, which is
given by [Neugebauer, 1975, p. 1313] for Massalia – 19 h-parallel, is not from Hipparchus but from
Strabo.
11 [Neugebauer, 1975, p. 304] states that the a-data form an arithmetical progression of the second
order. This is based, among others, on the value a = 3 c for M = 19 h. In F61 (G II.1.18), however,
Strabo says that the a belonging to M = 19 h is less than 3 c. An arithmetical progression is not
considered here.
10
Ptolemy gives the calculation of φ from M and vice versa by means of spherical
trigonometry in MS II.3. The comparison of Hipparchus’ M -data with the associated b0 -data by [Diller, 1934] and [Rawlins, 2009] suggests that Hipparchus used a
conversion φt (M,ε) based on spherical trigonometry too. The modern formulation of
Ptolemy’s computation of φ from M is
◦
φt (M,ε) = arctan (− cos (M/2 · 15 )/ tan ε)
h
(16)
(M in h), which is applied here. For ε the value εh is used, which probably underlies
Hipparchus’ conversion between M and φ (see [Diller, 1934], [Rawlins, 2009]). This
value has only an inconsiderably small difference to the actual value 23◦ 43′ in Hipparchus’ time. Since Hipparchus presumably converted M into φ and Strabo φ into
b0 , a conversion according to [Rawlins, 2009] is used here: φt (M ) is rounded to the
1 ◦
and b0 (φt ) to the nearest 100 st. The latter rounding is not applied to
nearest 12
the data of the “Commentary”.
For the conversion of ζa = 90◦ − δa into φ, equation (6) applies so that φ = ζa .
3.1
Notes on the data
The data on the Borysthenes refer to its mouth, cf. Section 2.1. In F57 (G II.5.42),
however, Strabo discusses “the regions in neighbourhood of the Borysthenes and the
southern parts of Lake Maeotis”. These regions are distinguished from the Borysthenes
and referred to as “Lake Maoetis” here.
F15 (G II.1.12): From this passage it follows that Meroë and the southern headlands of India have the same latitude. According to G II.1.20, however, this is objected
by Hipparchus in his second book (cf. [Berger, 1869, pp. 42, 97]) so that the information is not considered here.
F43 (G II.5.35): According to this passage, the Cinnamon country is situated “. . .
very nearly half-way between the equator and the summer tropic . . . .” Following
[Berger, 1869, p. 44], it is assumed here that this inaccurate localisation is not from
Hipparchus.
F46.3 (G II.5.36): [Diller, 1934] indicates that b0 = 11,800 st of Meroë, which
results from F43 (G II.5.35), is contradictory to b0 ≈ 11,600 st resulting from the
conversion of M = 13 h of the associated klima 12 in F46 (G II.5.36). [Rawlins, 2009]
points out the difference between the city Meroë and the Meroë-klima and gives b0 for
the klima. According to that, for bMk
of the Meroë-klima and bAk
0
0 of the AlexandriaMk
Mk
Mk
klima follows from F46: b0 + (b0 − 1,800 st) = bAk
= (bAk
0 , b0
0 + 1,800 st)/2. By
Mk
means of bAk
= 11,600 st.
0 = 21,400 st (see Section 3.2) it follows that b0
F47.3 (G II.5.36): “In Syene [. . . ] the sun stands in the zenith at the summer
solstice . . . .” Thus, as = 90◦ can be derived. From (12) follows φ = ε, for which
the actual value 23◦ 43′ is applied, since a real observation is assumed at first. Hence,
b0 = 16,602 st.
F48.4 (G II.5.38): Hipparchus distinguishes between Alexandria and the region
400 st south of it (Alexandria-klima, see F48.1) where M is 14 h (F48.3). For re of
12 The term klima denoted a latitudinal strip or a latitude which was assigned to a specific M value; in this regard see, e.g., [Honigmann, 1929], [Dicks, 1960, pp. 154–164], [Neugebauer, 1975, pp.
725–727].
11
Alexandria specified by Strabo (F48.4), [Jones, 1932, p. 511] and [Dicks, 1960, p. 95]
give 35 , which is an alteration. The correct value of the text is 57 (cf. [Neugebauer, 1975,
p. 336], [Rawlins, 2009]). Since this value yields a totally wrong φ, it is assumed (cf.
[Neugebauer, 1975, p. 336]) that it is an m : M -ratio of the length m of the shortest
day to the length M of the longest day, which was a usual quantity for the specification
of the latitude. From m = 24 h − M follows M = 24 h/(m/M + 1) = 14 h, which is
used here. Thus, contradictory to the text, this value does not refer to Alexandria
but to the Alexandria-klima (likewise [Rawlins, 2009]).
F48.6 (G II.5.38): Strabo gives re = 11
7 for Carthage, which yields the grossly erroneous φ = 32◦ 28′ by means of (14) (real φ = 36◦ 51′ ). [Rawlins, 1985, Rawlins, 2009]
assumes a similar error for Carthage as for the Alexandria-klima (see F48.4). According to this, the given re = 11
7 would be an M : m-ratio, which corresponds to
the common klima of M = 14 32 h (cf. [Neugebauer, 1975, p. 722]). This explanation
is not followed here. First, the ratio 57 for the Alexandria-klima is assumed to be
an m : M -ratio, but the ratio 11
7 for Carthage would be an M : m-ratio so that a
further inconsistency in the text would have to be assumed. Second, Ptolemy gives
φ = 32◦ 40′ and M = 14 51 h for Carthage (GH IV.3.7, VIII.14.5). Since Ptolemy’s
data rather originate from Hipparchus’ data than from Strabo’s data,13 Hipparchus’
φ must be about 32◦ 40′ , which is fulfilled by re .
F52.2 (G II.5.41): In F52 the ratio rs = 120/41 54 is given for Byzantium. According to F53 (G I.4.4) Hipparchus found the same ratio in Byzantium as Pytheas in
Massalia. If rs is Hipparchus’ ratio for Byzantium, a real measurement by Hipparchus
is unlikely because the ratio yields an error of about 2◦ with respect to the actual φ.
[Jones, 2002], for example, assumes a calculative origin for rs and recalculates 1/rs
from M = 15 41 h (F52.1) by means of εh , εm and εr (cf. Sections 2.1, 2.4) but does
not achieve identicalness with 120/41 54 .14 Following [Rawlins, 2009], it is assumed
here that rs is the result of a real measurement which Pytheas performed in Massalia.
Hence, ε = 23◦ 44′ of Pytheas’ time is used for conversion (15).
F56 (G II.5.41): “If one sails into the Pontus [Black Sea] and proceeds about 1,400
stades [F56.1] northwards, the longest day becomes 15 21 equinoctial hours [F56.2].”
The distance refers to the parallel of Byzantium (cf. [Dicks, 1960, p. 183]). Furthermore, the mentioned region in the Pontus (Mid-Pontus) is “. . . equidistant from the
pole and the equator . . . ” so that b0 = C/8 = 31,500 st (cf. (1)) is used (F56.3).
Moreover, there “. . . the arctic circle is in the zenith . . . ” (i.e. only one point of the
circle). Hence, its declination δa equals φ and (6) yields φ = 90◦ /2 = 45◦ or b0 = C/8,
which is not introduced here once more.
F57 (G II.5.42): Strabo reports on the neighbourhood of the Borysthenes and the
13 For example, Ptolemy states (GH I.4.2) that altitudes of the pole originating from Hipparchus
were available to him.
14 In order to test the possibility of an ancient conversion from M to r , the conversion of
s
[Jones, 2002] is redone with differing calculation steps in the following. The original conversion
from M to φ was presumably based on εh ([Diller, 1934]). Its result was probably rounded to the
1 ◦
: φt (M ) = 43◦ 17′ ≈ 43◦ 15′ . The conversion from φ to rs is based on (15), i.e. on the
nearest 12
determination of tan (φ − ε) = tan α. As it can be expected from ancient calculations, the tangentfunction was determined by the ratio crd(2α)/ crd(180◦ − 2α) (following from 2 sin α = crd 2α, see
[Neugebauer, 1975, pp. 21–24]). The chord crd() is determined here by a linear interpolation of
Hipparchus’ presumable table of chords, which was reconstructed by [Toomer, 1974]. The results for
εh , εm , εr are ≈ 42.69/120, ≈ 42.25/120 and ≈ 41.91/120, respectively. The resulting numerator,
however, should be within the interval [41 3.5
= 41.7, 41 4.5
= 41.9].
5
5
12
southern parts of Lake Maeotis: “The northern part of the horizon, throughout almost
the whole of the summer nights, is dimly illuminated by the sun [. . . ]; for the summer
tropic is seven-twelfths of a zodiacal sign from the horizon [= at ], and therefore this is
also the distance that the sun is below the horizon at midnight [= α].” One zodiacal
7
of a sign is 17◦ 30′ . The text suggests that
sign corresponds to 360◦ /12 = 30◦ and 12
the angles at and α refer to the summer solstice, that at is 17◦ 30′ and that α equals
at . Figure 4 shows the meridian m, the equator e and the positions D and N of the
observer at noon and at midnight, respectively, at the summer (a) and winter (b)
solstice. Shifting the horizon hD in the centre C of the earth to position h′D , m can be
regarded as the celestial sphere. Then, the sun altitude as at C in Figure (a) and the
angle αw at C in Figure (b) correspond to Strabo’s description of at and αs at N in
Figure (a) and αw at N in Figure (b) correspond to Strabo’s description of α. Owing
to αs 6= as , Strabo’s equation α = at holds true for the winter solstice only. For the
summer solstice, as = 90◦ − φ + ε ≈ 64◦ 57′ (equation (12) with b0 = 34,100 st of Lake
Maeotis, cf. Section 3.2) applies, which is inconsistent with the given value 17◦ 30′ of
at . The altitude aw = 90◦ − φ − ε ≈ 17◦ 37′ (equation (13)) of the winter solstice,
however, is consistent with the value of at but inconsistent with Strabo’s description
of at . From Figure (a) follows αs = 90◦ − σs = 90◦ − φ − ε = aw (equation (13)); from
Figure (b) follows αw = 90◦ − σw = 90◦ − φ + ε = as (equation (12)). In summary,
Strabo’s information can be corrected by the following two statements. First, at is
(about) 65◦ at the winter solstice and equals α at the winter solstice (= αw ). Second,
aw is 17◦ 30′ and equals α at the summer solstice (= αs ). Nonetheless, Strabo’s
information is not used further.
F60.1 (G II.5.42): [Gosselin, 1798, p. 28]15 , [Berger, 1869, p. 70] and [Diller, 1934]
notice that b = 6,300 st for Byzantium – “north of Lake Maeotis” is 1,400 st (2◦ )
too small in comparison to the M -data of these locations (F52.1: 15 41 h, F60.3: 17 h).
[Diller, 1934] assumes that Strabo inadvertently used Mid-Pontus (F56.2: M = 15 21 h)
instead of Byzantium for the calculation of b. Accordingly, the corrected b = 7,700 st
is used here.
F61.1 (G II.1.18): [Dicks, 1960, p. 185] shows that b = 6,300 st of Massalia –
Celtica (according to Hipparchus, but according to Strabo north of Celtica) has also
an error of 2◦ as b of F60.1. It is corrected to 7,700 st here.
F61.3 (G II.1.18): b = 9,100 st of Massalia – 18 h-region has the same error of 2◦ as
b of F60.1 ([Gosselin, 1798, p. 28], [Berger, 1869, p. 70], [Diller, 1934]). The corrected
b = 10,500 st is applied here.
F63 (CP I.3.7): Hipparchus refers to regions at the Hellespont. These regions
are equated here with Alexandria in the Troad. Hipparchus gives M : m = 53 and
M = 15 h. Since both are equivalent, only the latter is used here.
F65.1 (CP I.4.8): Hipparchus says that ζa is about 37◦ in the environment of
Athens and there where re = 34 . Although he does not explicitly assign this re to
Athens, it is used here for Athens.
F66.1, F67.1, F71.1 (CP I.7.11, I.7.14, II.4.2): Hipparchus gives the same data
of the culmination, rising and setting of constellations for Greece as for the regions
where M = 14 21 h. Thus, this M -value is assigned to Greece here.
15 I thank one of the referees for his reference to [Gosselin, 1798] in the context of the errors of
F60.1 and F61.3.
13
3.2
Test of consistency
The consistency of the data ascribed to Hipparchus is tested according to Section 2.2.
The n = 84 data bi given in Table 2 are composed to the equation system (4). The
imprecise aw of F61.5 is not involved in the adjustment computation. Furthermore,
the M -data of F66.1, F67.1 and F71.1 are not used because probably they are imprecise values (see below). There remain n = 80 data for the adjustment computation.
The equation system (4) has 34 unknown b0 of the locations shown in Figure 3 (for
Celtica only one b0 is used). The L1 -norm adjustment by means of the BR-algorithm
yields 61 vi being 0; hence, the related data are consistent among each other. The
vi 6= 0 are considered in the following. It turns out that the solution for the b0 can
be regarded as to be in accord with Hipparchus’ original data. The b0 are given in
Figure 3.
F15.1; Meroë – Byzantium; b ≈ 18,000 st; v = 500 st: 18,000 st are contradictory
to b = 18,500 st which follows from b0 of F43.2 (Cinnamon country), b0 of F52.3
(Byzantium) and b of F43.2/44.1 (Cinnamon country – Meroë). Since, however,
Strabo gives “about” 18,000 st, there is not a real contradiction.
F47.3; Syene; sun at zenith at summer solstice ⇒ b0 = 16,602 st; v = 198 st: b0
was derived from φ = 23◦ 43′ = ε (Section 3.1); it is contradictory to b0 = 16,800 st
which follows from b0 of F43.2 (Cinnamon country), b of F43.1 (Cinnamon country
– Meroë) and b of F43.3 (Meroë – Syene). 16800 st corresponds to φ = 24◦ in good
agreement with the real φ = 24◦ 05′ . Thereby, it equals the common ancient value εr
so that Syene was theoretically located on the tropic. Nonetheless, the information
of F47.3 may be based on a real observation. Owing to the closeness to the tropic,
the sun altitude has been about 90◦ at noon at the summer solstice so that for an
observer the sun apparently stood in the zenith.16
F48.6; Carthage; re = 11
7 ⇒ b0 = 22,730 st; v = −30 st: b0 of F48.3 (Alexandriaklima) and b of F48.5 (Alexandria-klima – Carthage) as well as M of F49.2 (Ptolemais
in Phoenicia) and b of F49.4 (Carthage – Ptolemais) yield b0 = 22,700 st for Carthage.
If this value was calculated from φ and rounded to the nearest 100 st, the original b0
should be within the interval [22,650 st, 22,750 st]. This is fulfilled by the b0 derived
from re . Hence, re could have been calculated from φ.
F50.5; Alexandria – Rhodes (centre); b = 3,640 st; v = −40 st: b is contradictory
to the value 3,600 st which follows from b = 7,000 st of F51.5 (Alexandria – Alexandria in the Troad) and b = 3,400 st of F51.7 (Rhodes – Alexandria in the Troad).
[Diller, 1934] assumes an error of 40 st in b = 3,640 st originating from a faulty reading. [Berger, 1869, p. 53] states that the value of 3,640 st is given with a higher
precision and that it refers to the city Rhodes. [Dicks, 1960, p. 176] considers b to be
derived from a real measurement by Hipparchus in the centre of Rhodes and assumes
that the text originally gave 3440 st. [Shcheglov, 2007] assumes that both 3,600 st
and 3,640 st are authentic and that they refer to the centre of Rhodes and the city
Rhodes, respectively. The following explanation shall be added. In F50 Strabo refers
not only to the centre of Rhodes but to “. . . the regions round the centre of Rhodes
. . . ” and states that there M is 14 21 h. Hipparchus assigned a φ-value to the 14 21 hklima, which probably was φ1 = φt (M ) = 36◦ 15′ . Strabo converted φ1 into b0 and
16 For 220–120 BC, the maximal sun altitude at the summer solstice was determined based on the
calculation method given in Fn. 5; the result is 89◦ 37′ ≈ 90◦ .
14
rounded it to the nearest 100 st: b01 (φ1 ) = 25,375 st ≈ 25,400 st. Strabo had a further
value φ2 for the city Rhodes in the north of the island (e.g., from Hipparchus, who
lived on the island Rhodes); it corresponded to b02 = 25,440 st (= 3640 st+ 21800 st of
Alexandria), i.e. φ2 ≈ 36◦ 21′ . (That value is somewhat less than the real φ = 36◦ 26′
and therefore consistent with the ancient systematic error of a gnomon measurement
due to the generation of the shadow by the upper edge of the sun, see Section 3.)
Strabo knew that M ≈ 14 21 h is spaciously valid, that φ1 is a theoretical value derived
from M and that his rounding up of b01 yielded a less accurate and more northerly
position (as b02 ). Furthermore, b01 and b02 only differ by 40 st. Thus, Strabo chose
the more precise and trustable b02 for his statement in F50 on the 14 21 h-klima.
F52.2; Byzantium; rs = 120/41 54 ⇒ b0 = 30,243 st; v = 57 st: rs is assumed to be
the result of an independent measurement (cf. Section 3.1); nevertheless, the small v
shows that rs is in accord with the other data.
F56.3; Mid-Pontus; b0 = C/8 ⇒ b0 = 31,500 st; v = 200 st: b is contradictory to
b0 = 31700 st which follows, for instance, from M of Mid-Pontus (F56.2). However, v
is acceptable because b0 of F56.3 is derived from rough information (cf. Section 3.1).
F57.4; Lake Maeotis; aw = 9 c ⇒ b0 = 33,833 st; v = 267 st: Since aw is a rounded
value, it can be regarded as consistent with the other data if v is < 0.5 c = 1◦ =
ˆ 700 st.
That is fulfilled.
F58.2; Borysthenes; aw = 9 c ⇒ b0 = 33,833 st; v = 167 st: Cf. F57.4.
F60.2; “north of Lake Maeotis”; aw = 6 c ⇒ b0 = 38,033 st; v = −33 st: Cf. F57.4.
aw is consistent with b0 = 38,000 st which follows from b0 of F52.4 (Byzantium) and
the corrected b = 7,700 st of F60.1 (Byzantium – “north of Lake Maeotis”).
F61.1; Massalia – Celtica; b = 7,700 st; v = −4,000 st: b is contradictory to the
value 3,700 st which follows from b of F59.2 (Massalia – Borysthenes) and b = 0
of F58.1/59.3 (Borysthenes – Celtica). From F61.1 and b0 = 30,300 st of Massalia
follows b0 = 38,000 st for Celtica in contrast to b0 = 34,000 st which follows from
F59.2 and F58.1/59.3. This is not a real contradiction because Celtica is a region
with a large latitudinal extent and Hipparchus did not distinguish between the Celtic
and the Germanic coast (see [Dicks, 1960, pp. 185, 188]) so that he gave a southern
(F58.1, F59.3) and a northern (F61.1, F61.2) latitude for Celtica. This becomes
evident by Strabo’s statement (F61/G II.1.18) that Hipparchus takes the inhabitants
of the region concerning F61.1 “. . . to be still Celts . . . ” and that Strabo himself
considers them as “. . . Britons who live 2,500 stades north of Celtica . . . ” (“Celtica”
refers to Hipparchus’ southern latitude). Strabo’s b = 2,500 st must be corrected by
+2◦ =
b 1,400 st as b of F61.1 (see Section 3.1). Then, for the northern latitude of
Celtica b0 = (34,000 + 2,500 + 1,400) st = 37,900 st is obtained in accordance with
38,000 st derived from F61.1.
F61.2; Celtica; aw = 6 c ⇒ b0 = 38,033 st; v = −4,033 st: As F61.1 (Massalia –
Celtica), aw refers to the northern latitude of Celtica at b0 = 38,000 st. The v in this
regard is only −33 st, which is acceptable, cf. F57.4.
F61.4; 18 h-region; aw = 4 c ⇒ b0 = 40,833 st; v = −33 st: Cf. F57.4.
F61.5; “inhabited region”; aw < 3 c ⇒ b0 > 42,198 st; v = 567 st: The information
b0 > 42,198 st is in accord with v > 0 and v is acceptable because of the imprecise
data; therefore, F61.5 is consistent.
F62.1, F65.1; Greece, Athens; re = 34 ⇒ b0 = 25,809 st; v = 91 st: From re and
(14) follows φ = 36◦ 52′ . Thus, re is consistent because it is in accord with the latitude
15
of 37◦ which follows from F62.3, F69.1 (Greece) as well as from F64.2, F65.2, F68.1,
F70.1 (Athens).
F62.2, F64.1; Greece, Athens; M = 14 53 h ⇒ b0 = 26,024 st; v = −124 st:
φt (M,εh ) = 37◦ 18′ . The difference to 37◦ of F62.3, F69.1 (Greece) as well as of
F64.2, F65.2/70.1, F68.1 (Athens) is 18′ . [Shcheglov, 2007] considers 37◦ as inconsistent with εh . φt (M,ε = 23◦ 51′ ) = 37◦ 03′ is in better agreement so that [Dicks, 1960,
p. 167] assumes 23◦ 51′ (MS I.12) to be Hipparchus’ value for ε. However, εh need not
be refused. First, Hipparchus only gives “about 37◦ ” in F62.3 (Greece) as well as in
F64.2, F65.2, F70.1 (Athens). Second, Hipparchus usually uses a step width of 41 h or
a multiple of it for his klimata; the nearest M -values 14 21 h and 14 43 h yield 36◦ 15′ and
38◦ 47′ so that 14 53 h represents a good fit with 37◦ and can be regarded as consistent.
Moreover, Hipparchus assigns 14 43 h (F62.2) as well as 14 21 h (e.g., F66.1) to Greece,
which illustrates Hipparchus’ low demand for the accuracy of the M -data.
F63.1; Alexandria in the Troad; M = 15 h ⇒ b0 = 28,753 st; v = 47 st: M is
consistent, it is in agreement with F51.4.
F63.2; Alexandria in the Troad; φ = ap ≈ 41◦ ⇒ b0 ≈ 28,700 st; v = 100 st: v is
acceptable because of the approximate ap so that F63.2 is consistent.
F66.1, F67.1, F71.1; Greece; M = 14 21 h ⇒ b0 = 25,308 st; v = 592 st: M is
contradictory to M = 14 53 h of F62.2. The smaller M = 14 21 h leads to a region south
of Athens because Hipparchus assigns Athens to M = 14 53 h (F64.1). Hipparchus
probably only gives a less accurate M -value with a resolution of 21 h in F66.1, F67.1,
F71.1.
F70.2; Rhodes; ζa = 36◦ ⇒ b0 = 25,200 st; v = 200 st: From M = 14 21 h of F50.4
follows φt (M ) = 36◦ 15′ . Hence, ζa of F70.2 is probably only a rough value as ap of
Athens of F64.2.
Since the aw -data of F60.2, F61.2 and F61.4 are inconsistent with the uncorrected
textual b-values of F60.1, F61.1 and F61.3 (see Section 3.1), they confirm that the
error of 1400 st of these b-values is caused by Strabo.
In F51 Strabo assigns Alexandria in the Troad to the parallel which has M = 15 h
(F51.4) and is “over 28,800 st” from the equator (F51.6). From M follows φt (M ) ≈
◦
40 61 =
b 28,817 st ≈ 28,800 st (which corresponds to the result of the adjustment). In
his statement Strabo possibly refers to the value of 28,817 st, which resulted from his
conversion of φ into b0 .
According to F53 the parallel through the mouth of the Borysthenes runs through
Britain too (F53.1). It is likely that Hipparchus referred to Celtica and Strabo replaced it by Britain (likewise [Berger, 1869, p. 66, fn. 1]) for the following reasons.
First, according to F58.1/59.3 (G II.1.18/12) Hipparchus locates Celtica and the Borysthenes at the same latitude. Second, according to F61 (G II.1.18) Hipparchus locates Britain north of the “inhabited region”/19 h-parallel, i.e. much more northernly
than the Borysthenes. Last, according to F61 Strabo believes the Celts mentioned by
Hipparchus to be Britons.
[Dicks, 1960, p. 184] assumes that Strabo’s data on the Borysthenes including F57
always refer to its mouth. There is, however, evidence that Hipparchus located the
mouth of the Borysthenes further south than the neighbouring region of the 16 h-klima
(F57). Strabo explicitly states that Hipparchus locates the mouth 3,700 st north of
Massalia and Byzantium (F59.1, F59.2) and 34,000 st north of the equator (F59.4);
from the former value also follows b0 = 34,000 st. Furthermore Strabo says that M
16
is 16 h in the regions in neighbourhood of the Borysthenes and the southern parts of
Lake Maeotis (F57.2), which are 3,800 st north of Byzantium (F57.1) and 34,100 st
north of the equator (F57.3); from the former b-value also follows b0 = 34,100 st. The
9 ◦
=
b 34125 st ≈ 34,100 st. Hence, Hipparchus
b-data are confirmed by φt (M ) ≈ 48 12
distinguished between the mouth of the Borysthenes and the 16 h-klima, which is
100 st further north.
4
Summary
The latitudinal data attributed to Eratosthenes and Hipparchus were each compiled
and formulated as systems of equations, whose solution revealed the differences and
inconsistencies of the data. As a result, the presumably original data of Eratosthenes
and Hipparchus were deduced.
The analysis of the data ascribed to Eratosthenes showed several disagreements,
which suggests that the concerned data originate from Strabo and not from Eratosthenes; this applies to F34.9, F34.10, F34.11, F35.5, F35.6, F36.2 and F47.1. In
particular, Eratosthenes’ latitudinal extent of the inhabited world up to the “northern regions” (F30.1, F34.12) is contradictory to the corresponding sum of the given
meridian arc lengths ascribed to Eratosthenes so far. Therefore, Eratosthenes’ meridian arc length of the part Borysthenes – “northern regions” is probably not 4,000 st
(F34.9) but 3,000 st, which is given by Strabo in G II.5.8 (F34.13).
Eratosthenes’ latitudinal distances Alexandria – Hellespont – Borysthenes (F35.2,
F35.3/36.3) are grossly erroneous. According to Strabo it was generally agreed that
the sea route Alexandria – Borysthenes is a straight line. Hence, Eratosthenes presumably based his latitudinal distances Rhodes – Hellespont – Byzantium on the
lengths of sea routes, which is affirmed by a good agreement of his distances with the
actual distances alongside the Turkish coast.
From Pytheas’ information on the position of the arctic circle relating to Thule it
was known that Thule is situated at a latitude of (90◦ − ε), where ε is the obliquity of
the ecliptic. In conjunction with Eratosthenes’ latitudinal data for Thule, 23◦ 40′ can
be derived for Eratosthenes’ value of ε. This value corresponds to Hipparchus’ presumable value (see [Diller, 1934]) and was possibly referred to by Ptolemy in MS I.12.
The fragments ascribed to Hipparchus contain latitudinal quantities of different
types. Occurring differences of the data were explained by the different types of information and their different precision and origination. The real inconsistencies can
be ascribed to Strabo in most cases; this applies to F48.4 (e.g., [Neugebauer, 1975]),
F60.1 and F61.3 ([Diller, 1934]), F61.1 ([Dicks, 1960]), F15.1, F51.6, F53.1 and F57.
Strabo’s statement on the distances of the summer tropic and the sun with respect to
the horizon at Lake Maeotis in F57 has not been interpreted so far; his error in this
regard was illustrated. Hipparchus distinguished between the 14 h-klima and the city
Alexandria as well as between the 13 h-klima and the city Meroë ([Rawlins, 2009]).
The present investigation revealed that Hipparchus probably also distinguished between the 16 h-klima and the mouth of the Borysthenes 100 st south of the parallel of
the klima.
17
Appendix: On the location of Thule
Pytheas’ voyage to Thule took place in ca. 330 BC.17 His treatise “On the Ocean”
on his voyages is not preserved, but later ancient authors provided extractions of this
treatise. The handed down information on Thule is given by, e.g., [Hennig, 1944,
pp. 155–159] and [Whitaker, 1982]. The main sources are Strabo’s “Geography” and
Pliny’s “Natural History”. The only quotation from Pytheas’ treatise is to be found
in Geminus’ Eisagoge (E; see [Manitius, 1898]). Furthermore, Ptolemy describes the
position and form of the island Thule by means of longitudes and latitudes in GH II.3.
The two common localisations of Pytheas’ Thule are Iceland (e.g., [Burton, 1875],
[Roller, 2010, p. 127]) and Norway. Iceland is neglected here because Pytheas met
inhabitants in Thule according to E VI.9, but so far a settlement of Iceland can not be
assumed for his time. [Nansen, 1911, p. 62] and [Hennig, 1944, p. 166] locate Thule
in the region of Trondheim in Norway.
Ptolemy does not refer to Pytheas’ Thule; his island Thule is usually identified
as Shetland (e.g., [Rivet and Smith, 1979, p. 146]). Detailed reasons for this are
given by [Marx, 2013b] in an investigation of Ptolemy’s coordinates of Scotland. It
should be added that furthermore Ptolemy’s length of the longest day in Thule of 20 h
(GH VIII.3.3, MS II.6) contradicts Pytheas’ information on the length of the nights
in Thule (see below).
For a localisation of Pytheas’ Thule, the following information comes into consideration:
1. Thule is a six-day seafaring from Britain in northern direction (G I.4.2, NH II.77).
2. In the region of Thule the tropic of Cancer coincides with the arctic circle (G II.5.8).
At the summer solstice there are no nights (NH IV.30).
3. The meridian arc length b of Borysthenes – Thule is 11,500 st (G I.4.2).
4. Pytheas said that in Thule the length n of the nights was 2 h and 3 h (E VI.9).
5. A one-day journey from Thule is the frozen/clotted sea (NH IV.30).
The frozen/clotted sea suggests a larger appearance of sea ice. That disagrees
with the location of Thule in Norway because in the Norwegian Sea there is no drift
ice (cf. [Vinje and Kvambekk, 1991]) and at Pytheas’ time, at the beginning of the
Subatlantic, the climate was similar to today’s climate so that drift ice can be excluded. According to [Hennig, 1944, pp. 105, 156, fn. 1] the clotted sea is a fiction,
which can be found similarly in ancient and medieval literature. Thus, information 5
does not play a role here.
Information 2 and 3 are treated in Section 2.4; information 1 and 4 are dealt with
in the following.
For the conversion of the time of six days (i.e. days and nights) into a length of
a sea route, there is the following information. First, Strabo says (G X.4.5) that
the sea voyage from Cyrenaea (Cape Rasat) to Criumetopon (Cape Krio) takes two
days and nights (see Fn. 6); from this distance of 313 km a speed of 156.5 km/d
follows. Second, Eratosthenes probably assessed 1000 st for a day-and-night-seafaring
(see Fn. 6); by means of the Egyptian stadium (cf. Section 2.3) a speed of 157.5 km/d
results. Both results are similar and yield a length of about 940 km for the sea
route of Pytheas’ voyage, which is used here. According to Pliny (NH IV.30) one
traveled from the island Berrice (also named Nerigos in the manuscripts) to Thule.
17 [Nansen,
1911, p. 48]: 330–325 BC; [Hennig, 1944, p. 162]: 350–310 BC.
18
Berrice is possibly the island Mainland of Shetland, cf. [Nansen, 1911, p. 61] and
[Hennig, 1944, p. 156]. The starting point of the six-day journey to Thule, however,
was rather located at Great Britain, since Thule “. . . is six days’ sail from the north
of Britain . . . ” ([Bostock and Riley, 1855]) according to NH II.77 ([Dilke, 1985, p.
136], for example, chooses Cape Wrath for the starting point). NH IV.30 suggests
that Berrice/Mainland of Shetland was on Pytheas’ way to Thule, which is taken into
account here. For the starting point of the time measurement of the six-days journey
Duncansby Head is assumed here. Figure 5 shows two possible sea routes with a length
of 940 km from Great Britain to Thule. Both routes bypass Orkney and Mainland.
From there, route A takes course directly to the West Cape of Norway and continues
alongside the Norwegian coast up into the Trondheimsfjord. Route B takes course
eastwards along a constant latitude to the Norwegian coast at Bergen and continues
alongside the coast up to the island Smøla. Possibly, such a latitude sailing was used,
which was an easy and common method for navigation (it was used, e.g., by the
Vikings later on, cf. [Johnson, 1994]). [Hennig, 1944, p. 167] rejects the similar route
Orkney – Bergen because in his opinion the eastern course contradicts the position of
Thule north of Britain. If, however, Pytheas visited a northern region of Norway and
only referred the name Thule to this region, then there is no contradiction. Owing
to the uncertainty of the assumed speed and way, Pytheas’ landing point can not
be given precisely. The Trondheimsfjord and the coastal region of the same latitude
come into consideration.
Before Geminus quotes Pytheas in E VI.9 he discusses M -data of different latitudes. Thus, [Nansen, 1911, p. 57] assumes that Pytheas’ information on the length
of the nights refers to the shortest nights of the year. This, however, does not result
directly from the text. For Pytheas’ time the lengths of the nights were determined;
the year 330 BC was used, other supposable times do not yield significant differences. The shortest nights (at summer solstice) with lengths N = 2 h,3 h occurred
at φ = 64◦ 40′ and φ = 63◦ 40′ , respectively.18 These latitudes are significantly less
than φ = 66◦ 16′ of the polar circle and thus inconsistent with the information on the
arctic circle (see Section 2.4), see also Figure 5. In Geminus’ quotation it is only said
that the night was very short so that the given lengths of nights may refer to a date
near the summer solstice. In order to locate the associated region, the length n of the
night was determined for different latitudes φ and times t. Figure 6 shows the result
in form of isolines of n. At φ = 63◦ 20′ n was 3.3 h ≈ 3 h five days before/after the
summer solstice. Thus, the southern limit for Pytheas’ Thule can be located at this
latitude, which corresponds to the latitude of the southern end of the Trondheimsfjord. At the polar circle, at φ = 66◦ 16′ , n was ≈ 2 h 22 days before/after the summer
solstice. Possibly, Pytheas traveled to this region at that time, where he heard about
the midnight sun.
Pytheas’ information on the arctic circle and Eratosthenes latitudinal data lead
to the northern polar circle at φ = 66◦ 16′ at Pytheas’ time. Pytheas’ information
on the journey length suggests the region at the latitude of the southern end of the
18 For a location of latitude φ and for a given time t, the length n of the night can be determined
by the following calculation steps (according to [Strous, 2012]; applied formulas see [Meeus, 1991,
pp. 98, 135, 151–153]; cf. also Fn. 5): ε = ε(t); M (t); L0 = L0 (t); C = C(t,M ); Θ = Θ(L0 , C);
δ = δ(Θ, ε); hour angle at sunset: H = H(φ, δ, a0 ); n = 24 h − 2H. By means of the altitude
a0 = −50′ the atmospheric refraction and the size of the sun disc are taken into account. For the
determination of φ, it was varied till n equaled the given value.
19
Trondheimsfjord. His information on the length of the nights leads to both of these
regions. This is not contradictory because the name Thule may refer to a region
of larger extent. Hence, Pytheas’ Thule can be equated with the region of Norway
west of the Scandinavian Mountains between about 63◦ 20′ and 66◦ 16′ latitude. This
result is in accordance with Pytheas’ contact with inhabitants and his report on the
cultivation of grain in Thule (G IV.5.5). In the said region there are spacious lowlying areas and a warm and humid climate influenced by the North Atlantic Current.
Apart from the southern regions at the Skagerrak, in Norway there are only low-lying
areas with fertile clayey soils at the Trondheimsfjord (see [Sporrong, 2008, p. 26, fig.
6]). According to [Helle, 2008, pp. 7–8] there were stable settlements and farming
in Norway as far north as Trøndelag at the beginning of the Iron Age (500 BC –
800 AD). Furthermore, in regions at the polar circle agriculture was introduced in
the 7th and 4th century BC as revealed by radiocarbon dating based on pollen (see
[Johansen and Vorren, 1986]).
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22
Table 1: Latitudinal data of the western Oikoumene derived from the fragments (F;
with the exception of F34.13 from [Roller, 2010]) ascribed to Eratosthenes; ∗ = see
note
F
Source
From
To
b [st]
30.1, 34.12 G II.5.6,
Cinnamon country
northern regions
<30,000
II.5.9
30.2, 34.6
G II.5.6,
equator
Cinnamon country
8,800
II.5.7
≈
34.1, 35.1
G II.5.7,
Meroë
Alexandria
= 10,000
I.4.2
34.2
G II.5.7
Meroë
Syene
5,000
34.3
G II.5.7
Cinnamon country
Meroë
≈3,000
34.4
G II.5.7
Cinnamon country
Syene
8,000
34.5∗,
G II.5.7,
equator
Syene
16,800
M6.1∗
C I.7
34.7
G II.5.7
equator
Alexandria
21,800
34.8
G II.5.8
Byzantium
Borysthenes
≈3,800
34.9
G II.5.9
Borysthenes
northern regions
4,000
34.10
G II.5.9
Rhodes
northern regions
12,700
34.11
G II.5.9
Cinnamon country
Rhodes
16,600
34.13
G II.5.8
Borysthenes
northern regions
3,000
35.2
G I.4.2
Alexandria
Hellespont
≈8,100
35.3, 36.3
G I.4.2,
Hellespont
Borysthenes
5,000
II.5.42
35.4
G I.4.2
Borysthenes
Thule
≈11,500
35.5
G I.4.2
Cinnamon country Meroë
3,400
(, Egyptian island,
Taprobane)
35.6
G I.4.2
Cinnamon country
Thule
38,000
36.1
G II.5.42 Meroë
Borysthenes
&23,000
36.2, 47.1
G II.5.42, Meroë
Hellespont
18,000
G II.1.3
40+41∗
G II.1.20, Meroë
country of the Tro0
NH II.185
godytes
56.1
G II.1.33 Alexandria
Rhodes
.4,000
128.1
G II.5.24 Alexandria
Rhodes
3,750
23
Table 2: Latitudinal data derived from the fragments (F; with the exception of F62.1–
71.1 from [Dicks, 1960]) ascribed to Hipparchus; n. = north of, s. = south of
F
15.1
43.1
Source
G II.1.12
G II.5.35
From
Meroe
Cinnamon country
equator
43.2, 44.1
43.3
46.1
46.2
46.3
47.1
47.2
G II.5.35,
II.1.13
G II.5.35
G II.5.36
G II.5.36
G II.5.36
G II.5.36
G II.5.36
47.3
47.4
48.1
48.2
48.3
48.4
48.5
48.6
49.1
G II.5.36
G II.5.36
G II.5.38
G II.5.38
G II.5.38
G II.5.38
G II.5.38
G II.5.38
G II.5.35
49.2
G II.5.35
equator
equator
Alexandria-klima
Alexandria
equator
equator
Alexandria-klima
equator
Ptolemais
(Phoenicia)
equator
49.3
G II.5.35
Alexandria
49.4
G II.5.35
Carthage
50.1
50.2
50.3
50.4
50.5
51.1
G II.5.39
G II.5.39
G II.5.39
G II.5.39
G II.5.39
G II.5.40
51.2
G II.5.40
51.3
G II.5.40
51.4
G II.5.40
Rhodes
Rhodes
Rhodes
equator
Alexandria
Alexandria in the
Troad
Alexandria in the
Troad
Alexandria in the
Troad
equator
51.5
G II.5.40
Alexandria
Meroë
Meroë-klima
equator
equator
Syene
Syene
24
To
Byzantium
Meroë
b [st]
≈18,000
3,000
Cinnamon country
Syene
Ptolemais
Meroë-klima
Meroë-klima
Berenice
country of the
Trogodytes
Syene
Syene
Alexandria
Cyrene
Alexandria-klima
Alexandria-klima
Carthage
Carthage
Sidon/Tyre
=
≈ 8,800
5,000
0
11,600
11,600
0
0
16,602
16,800
≈400
0
21,400
21,400
1,300
22,730
0
Ptolemais
(Phoenicia)
Ptolemais
(Phoenicia)
Ptolemais
(Phoenicia)
Peloponnese
Xanthus
Syracuse
Rhodes
Rhodes
Amphipolis
23,400
Apollonia
0
s.
Rome & n.
Naples
Alexandria in the
Troad
Alexandria in the
Troad
0
Original
M = 13 h
see note
see note
M = 13 21 h
M = 14 h
see note
re =
11
7
M = 14 41 h
≈1,600
≈700
0
0
400
25,400
3,640
0
28,800
7,000
M = 14 21 h
M = 15 h
Table 2: Latitudinal data derived from the fragments (F; with the exception of F62.1–
71.1 from [Dicks, 1960]) ascribed to Hipparchus; n. = north of, s. = south of
F
51.6
Source
G II.5.40
From
equator
51.7
G II.5.40
Rhodes
51.8
G II.5.40
51.9
51.10,
53.2, 54.1,
55.1
G II.5.40
G II.5.40,
I.4.4,
II.5.8,
II.1.12
G II.5.41
G II.5.41
G II.5.41
G II.5.41
G I.4.4
G II.5.41
G II.5.41
G II.5.41
G II.5.42
G II.5.42
G II.5.42
G II.5.42
G II.1.18,
II.1.12
G II.1.18
G II.1.12
G II.1.12
G II.1.13
G II.5.42
G II.5.42
G II.5.42
G II.1.18
G II.1.18
G II.1.18
G II.1.18
G II.1.18
G II.1.18
G II.1.18
CP 1.3.6
CP 1.3.6
CP 1.3.6
Alexandria in the
Troad
Byzantium
Byzantium
52.1
52.2
52.3
52.4
53.1
56.1
56.2
56.3
57.1
57.2
57.3
57.4
58.1, 59.3
58.2
59.1
59.2
59.4
60.1
60.2
60.3
61.1
61.2
61.3
61.4
61.5
61.6
61.7
62.1
62.2
62.3
To
Alexandria in the
Troad
Alexandria in the
Troad
Byzantium
b [st]
>28,800
Nicaea
Massalia
0
0
equator
equator
Rhodes
equator
Borysthenes
Byzantium
equator
equator
Byzantium
equator
equator
equator
Borysthenes
Byzantium
Byzantium
Byzantium
Byzantium
Britain(?)
Mid-Pontus
Mid-Pontus
Mid-Pontus
Lk. Maeotis
Lk. Maeotis
Lk. Maeotis
Lk. Maeotis
Celtica
30,300
30,243
≈4,900
≈30,300
0
≈1,400
31,700
31,500
≈3,800
34,100
34,100
33,833
0
M = 15 41 h
rs = 120/41 54
equator
Byzantium
Massalia
equator
Byzantium
equator
equator
Massalia
equator
Massalia
equator
equator
equator
equator
equator
equator
equator
Borysthenes
Borysthenes
Borysthenes
Borysthenes
n. Lk. Maeotis
n. Lk. Maeotis
n. Lk. Maeotis
Celtica
Celtica
18 h-region
18 h-region
inhabited region
inhabited region
18 h-region
Greece
Greece
Greece
33,833
3,700
3,700
34,000
7,700
38,033
38,000
7,700
38,033
10,500
40,833
>42,233
42,800
40,800
25,809
26,024
≈25,900
aw = 9 c
25
Original
3,400
1,500
M = 15 21 h
see note
M = 16 h
aw = 9 c
6,300 st
aw = 6 c
M = 17 h
6,300 st
aw = 6 c
9,100 st
aw = 4 c
aw < 3 c
M = 19 h
M = 18 h
re = 43
M = 14 53 h
ap ≈ 37◦
Table 2: Latitudinal data derived from the fragments (F; with the exception of F62.1–
71.1 from [Dicks, 1960]) ascribed to Hipparchus; n. = north of, s. = south of
F
63.1
Source
CP 1.3.7
From
equator
63.2
CP 1.3.7
equator
64.1
64.2
65.1
65.2, 70.1
CP I.3.12
CP I.3.12
CP I.4.8
CP I.4.8,
I.11.8
CP I.7.11,
I.7.14,
II.4.2
CP I.7.21
CP I.11.2
CP I.11.8
66.1, 67.1,
71.1
68.1
69.1
70.2
equator
equator
equator
equator
To
Alexandria in the
Troad
Alexandria in the
Troad
Athens
Athens
Athens
Athens
equator
equator
equator
equator
26
b [st]
28,753
Original
M = 15 h
≈28,700
ap ≈ 41◦
26,024
≈25,900
25,809
≈25,900
M = 14 53 h
ap ≈ 37◦
re = 43
ζa ≈ 37◦
Greece
25,308
M = 14 21 h
Athens
Greece
Rhodes
25,900
25,900
25,200
ζa = 37◦
ζa = 37◦
ζa = 36◦
Figure 1: Places located on Eratosthenes’ prime meridian (italic) and supposable sea
routes underlying Eratosthenes’ data (lines)
27
b0 [st]
46400
Thule
35.4
37900
northern regions
34.9,13
30.1,
34900
Borysthenes
34.12
34.8
31100
35.3,
36.3
29900
Byzantium
34.10
Hellespont
36.1
35.2
25550
Rhodes
56.1, 128.1
36.2,
47.1
21800
34.1,
34.5,
35.1
Syene
M6.1
34.2
34.4
Trogodytes
Meroe
40+41
16800
11800
8800
35.6
34.11
Alexandria
34.7
equator
30.2,
34.6
34.3, 35.5
Cinnamon country
Figure 2: Graph of the latitudinal data of the western Oikoumene ascribed to Eratosthenes; the vertical order of the locations gives the meridian arc length b0 from the
equator (not drawn to scale)
28
b0 [st] M [h]
42800 19
61.5,6
inhabited region
40800 18
61.4,7
18h-region
38000 17
61.3
Celtica 2
60.2,3
n. Maeotis
60.1
34100 16
Maeotis
34000
58.1, 59.3
Celtica 1
1
31700 15 /2
Borysthenes
Britain(?)
53.1
Mid-Pontus
59.1
1
30300 15 /4
Massalia
61.2
51.3
28800 15
s. Rome,
n. Naples
φ∼37° 14 /5
Athens
3
1
25400 14 /2
56.1
51.10,53.2,
Byzantium 51.9 Nicaea
54.1, 55.1
51.8
52.1,2,4
51.2
Amph.
Apoll.
Greece
64.1,2, 65.1,2,
68.1, 70.1
25800
62.1-3, 66.1,
67.1, 71.1, 69.1
Syracuse
Peloponnes
51.1
1
Xanthus
Sidon/Tyre
56.2,3
Alex.Troad
50.3
58.2,
59.4
51.4,6,
63.1,2
51.7 52.3
50.2
50.4, 70.2
Rhodes
50.1
23400 14 /4
57.2-4
57.1
59.2
61.1
50.5
49.2
49.1 Ptolemais (Ph.)
48.6
49.3
49.4
22700
Carthage
21800
Cyrene
equator
48.3,4
51.5
48.5
Alexandria
48.2
48.1
Alex.-klima
21400 14
47.1
1
16800 13 /2
Berenice
Trogodytes
47.2
Syene
43.3
11800
43.2,
44.1
47.3,4
15.1
46.2,3
Meroe
11600 13
Ptolemais
46.1
3
8800 12 /4
Meroe-klima
43.1
Cinnamon country
Figure 3: Graph of the latitudinal data ascribed to Hipparchus; the vertical order of
the locations gives the meridian arc length b0 from the equator (not drawn to scale;
s. = south of, n. = north of)
29
as
ss
N
hN
S
D
as
hD
t
e
f
fe
ss e
as C
m
e
hD'||hD
(a)
D
aw
hD
N
S
t
aw
sw
sw
fe
sw
e
a
f
e C wa
hN
w
m
hD'||hD
e
(b)
Figure 4: On Strabo’s information on Lake Maeotis in fragment F57, (a) summer
solstice, (b) winter solstice (e: equator, hD/N : horizon at noon/midnight, m: meridian,
S: sun, t: tropic)
30
Figure 5: Pytheas’ possible sea route from Great Britain to Thule; route A: continuous, route B: dash-dot/continuous
66.0
65.5
1.5h
65.0
2.0h
φ [°]
64.5
2.5h
64.0
3.0h
63.5
3.5h
63.0
62.5
62.0
-40
-30
-20
-10
0
t [d]
10
20
30
40
Figure 6: Isolines of the length n of the night subject to the latitude φ and time t
expressed by the number of days since the summer solstice in 330 BC
31